II Escuela de Optica Biomedica, Puebla, 2011 Polarimeters Jessica C. Ramella-Roman, PhD

II Escuela de Optica Biomedica, Puebla, 2011 Stokes vector formalism Four measurable quantities (intensities) Characterize the polarization state of light E0x, E0y, Cartesian electric field component d=dx-dy phase difference

II Escuela de Optica Biomedica, Puebla, 2011 Simple Stokes vector polarimeter Six intensity measurements

II Escuela de Optica Biomedica, Puebla, 2011 Simple Stokes vector polarimeter Horizontal, i.e. parallel to reference frame polarizer

II Escuela de Optica Biomedica, Puebla, 2011 Simple Stokes vector polarimeter Vertical, i.e. perpendicular to reference frame polarizer

II Escuela de Optica Biomedica, Puebla, 2011 Simple Stokes vector polarimeter Linear polarizer at +/-45o to reference frame polarizer

II Escuela de Optica Biomedica, Puebla, 2011 Simple Stokes vector polarimeter Circularly polarized (left and right) Quarter-wave plate polarizer Polarizer and quarterwave plate axis are at 45 o to each other

II Escuela de Optica Biomedica, Puebla, 2011 A Mueller matrix polarimeter R1R2 P qwp P

II Escuela de Optica Biomedica, Puebla, measurements HH -> H source H detector HV -> H source V detector HP -> H source P detector HR -> H source R detector VH -> V source H detector VV -> V source V detector VP -> V source P detector VR -> V source R detector PH -> P source H detector PV -> P source V detector PP -> P source P detector PR -> P source R detector RH -> R source H detector RV -> R source V detector RP -> R source P detector RR -> R source R detector

II Escuela de Optica Biomedica, Puebla, measurements HH -> H source H detector HV -> H source V detector HP -> H source P detector HR -> H source R detector VH -> V source H detector VV -> V source V detector VP -> V source P detector VR -> V source R detector PH -> P source H detector PV -> P source V detector PP -> P source P detector PR -> P source R detector RH -> R source H detector RV -> R source V detector RP -> R source P detector RR -> R source R detector Handbook of optics Vol II

II Escuela de Optica Biomedica, Puebla, 2011 A Mueller matrix polarimeter R1R2 P qwp P

II Escuela de Optica Biomedica, Puebla, measurements HH -> H source H detector HV -> H source V detector HP -> H source P detector HR -> H source R detector VH -> V source H detector VV -> V source V detector VP -> V source P detector VR -> V source R detector PH -> P source H detector PV -> P source V detector PP -> P source P detector PR -> P source R detector RH -> R source H detector RV -> R source V detector RP -> R source P detector RR -> R source R detector

II Escuela de Optica Biomedica, Puebla, 2011 A Mueller matrix polarimeter R1R2 P qwp P

II Escuela de Optica Biomedica, Puebla, measurements HH -> H source H detector HV -> H source V detector HP -> H source P detector HR -> H source R detector VH -> V source H detector VV -> V source V detector VP -> V source P detector VR -> V source R detector PH -> P source H detector PV -> P source V detector PP -> P source P detector PR -> P source R detector RH -> R source H detector RV -> R source V detector RP -> R source P detector RR -> R source R detector

II Escuela de Optica Biomedica, Puebla, 2011 A Mueller matrix polarimeter R1R2 P qwp P

II Escuela de Optica Biomedica, Puebla, 2011 Special issues in polarimetry Spectral stokes vector optimization Mueller matrix optimization

II Escuela de Optica Biomedica, Puebla, 2011 Motivations Stokes vector polarimeter can be used for rough surface measurements characterization of particle size (partial Stokes vectors, co cross polarization) Multi-spectral Stokes vector polarimeters are costly, often we need to sacrifice spectral performance (single wavelengths)

II Escuela de Optica Biomedica, Puebla, 2011 Experimental Layout LCR1- Liquid Crystal Retarder  = 0 o LCR2- Liquid Crystal Retarder  = 45 o p polarizer Fiber – 200µm LED – White LED or Xenon wls p

II Escuela de Optica Biomedica, Puebla, 2011 Experimental Layout for Mueller M P We observe the spectrum between 550 and 750 nm LCR1- Liquid Crystal Retarder  = 0 o LCR2- Liquid Crystal Retarder  = 45 o p polarizer Fiber – 200µm LED – White LED or Xenon wls p WP

II Escuela de Optica Biomedica, Puebla, 2011 Calibration Method was originally proposed by Boulbry et al.* for an imaging system and 3 wavelengths. Calibration does not require ANY knowledge of LCR retardation or orientation There is a linear transformation between a set of measurements and the Stokes vector *B. Boulbry, J.C. Ramella-Roman, T.A. Germer, Applied Optics, 46, pp. 8533–8541, 2007.

II Escuela de Optica Biomedica, Puebla, 2011 Theta is the orientation angle of the polarizer with respect to the reference plane, 0 to 180 o Six spectra I i, are acquired for each theta for different LCR retardation p WP achromatic ¼ wave plate Polarizer after wave plate

II Escuela de Optica Biomedica, Puebla, 2011 Polarizer before wave plate Theta is the orientation angle of the polarizer with respect to the reference plane, 0 to 180 o Six spectra I i, are acquired for each theta for different LCR retardation p WP achromatic ¼ wave plate

II Escuela de Optica Biomedica, Puebla, 2011 Calibration cnt. The calibration polarizer and wave plate ideally create the Stokes vectors M Mueller matrices S Stokes vectors

II Escuela de Optica Biomedica, Puebla, 2011 Calibration cnt. The Stokes vectors are related to the measured values Ii through the data reduction matrix W for which W is finally calculated using the SVD of I

II Escuela de Optica Biomedica, Puebla, 2011 Calibration cnt. Once W is know only 6 I measurements are necessary to build the full Stokes vector This is true at every wavelength.

II Escuela de Optica Biomedica, Puebla, 2011 Results - Incident [ ] 90 o

II Escuela de Optica Biomedica, Puebla, 2011 Results - Incident [ ] 45 o wp

II Escuela de Optica Biomedica, Puebla, 2011 Is chicken a perfect wave-plate? Angle Wavelength Transmitted degree of polarization [ ] P

II Escuela de Optica Biomedica, Puebla, 2011 Chicken muscle ~ cylinder scattering + Rayleigh scattering OASIS 2011 DLP Real Simulated DCP

II Escuela de Optica Biomedica, Puebla, 2011 More on chicken and polarization on

II Escuela de Optica Biomedica, Puebla, 2011 The same layout & calibration can be used to build a Mueller matrix polarimeter 45 o wp

II Escuela de Optica Biomedica, Puebla, 2011 Mueller matrix of air 45 o wp

II Escuela de Optica Biomedica, Puebla, 2011 Mueller matrix of air 45 o wp

II Escuela de Optica Biomedica, Puebla, 2011 Conclusions Stokes vector polarimeter is fiber based and usable between nm Point measurements of small scatterers Miniaturizing the system

II Escuela de Optica Biomedica, Puebla, 2011 Optimization of Mueller Matrices measurements The classic Mueller matrix polarimeter Previous work on optimizing a polarimeter Mueller matrix polarimetry with SVD

II Escuela de Optica Biomedica, Puebla, 2011 Dual rotating retarder polarimeter R1R2 R1 : 5 R2 R1 R2

II Escuela de Optica Biomedica, Puebla, 2011 D. B. Chenault, J.L. Pezzaniti, R.A. Chipman, “Mueller matrix algorithms,” in D. Goldstein and R. Chipman (eds.), “Polarization analysis and measurement,” in Proc. Soc. Photo-Opt. Instrum. Eng. V. 1746, pp (1992) Measured flux Analyzing vector Source vector Sample Mueller matrix Calculation of Mueller Matrix

II Escuela de Optica Biomedica, Puebla, 2011 P q measured flux for q source detector retarders combination S q source vector (Stokes vectors of source polarizing elements) A q detectors vector (Stokes vectors of detector analyzing elements) Calculation of Mueller Matrix

II Escuela de Optica Biomedica, Puebla, 2011 Flattened Mueller Matrix Measurement matrix The q th measurement Calculation of Mueller Matrix aqsqaqsq

II Escuela de Optica Biomedica, Puebla, 2011 For 16 measurements W is square with a unique inverse – (if W non singular)

II Escuela de Optica Biomedica, Puebla, 2011 For more than 16 measurements W is not square so to calculate the Mueller matrix *M. Smith ‘Optimization of the dual-rotating-retarder Mueller matrix polarimeter”, Applied Optics, V. 41, No. 13 (2002)

II Escuela de Optica Biomedica, Puebla, 2011 Two main issues Which wave-plates are best for Mueller matrix calculation number of measurements to calculate the Mueller matrix

II Escuela de Optica Biomedica, Puebla, 2011 Which retarder are best for Mueller matrix calculation* *M. Smith ‘Optimization of the dual-rotating-retarder Mueller matrix polarimeter”, Applied Optics, V. 41, No. 13 (2002)  Change retardation of R1 and R2, (R1,R2 have same retardation)  200 measurements to calculate W  R1:R2, 1:5 ratio Calculate cond( W)

II Escuela de Optica Biomedica, Puebla, 2011 Condition number 1/cond(A) how close is A to a singular matrix

II Escuela de Optica Biomedica, Puebla, 2011 Best retardance 127   Minima at 127  and 233 

II Escuela de Optica Biomedica, Puebla, 2011 n of measurements* *M. Smith ‘Optimization of the dual-rotating-retarder Mueller matrix polarimeter”, Applied Optics, V. 41, No. 13 (2002)  Angular increments of source and detector retarders are varied  Angular increments 0:60  Fixed retardance 127 o  16 measurements  30 measurements Calculate cond( W)

II Escuela de Optica Biomedica, Puebla, 2011 Cond(W) Over-determined system are better 16 measurements 30 measurements

II Escuela de Optica Biomedica, Puebla, 2011 Svd vs pseudo-inverse 1Air 2 Linear P 3 qwp

II Escuela de Optica Biomedica, Puebla, 2011 Modeled error

II Escuela de Optica Biomedica, Puebla, 2011 Modeled error SVD Smith SVD gives low level error for broader range of retardances SVD pseudoinverse

II Escuela de Optica Biomedica, Puebla, 2011 Does the sample Mueller M bias results? 1Air 2 Linear P 3 qwp

II Escuela de Optica Biomedica, Puebla, 2011 Generating a sample Mueller matrix Generate 4 different Mueller matrices with 2,4,6,and 8 degrees of freedom (100 MM total) Check for physical plausibility of Mueller matrix (Handbook of Optics Vol II)

II Escuela de Optica Biomedica, Puebla, 2011 SVD Smith Error with reconstructed Mueller matrices SVD gives low level error for broader range of retardances SVD pseudoinverse

II Escuela de Optica Biomedica, Puebla, 2011 Conclusions Using SVD a broader range of retarders may be used in DRR polarimetry Several numerical programs (such as Matlab) use SVD in their pseudo-inverse algorithms In most cases the error due to use of SVD is minimal compared to instrumental errors

II Escuela de Optica Biomedica, Puebla, 2011 Tomorrow Monte Carlo modeling basics Monte Carlo with Meridian planes

II Escuela de Optica Biomedica, Puebla, 2011 Condition number 1/cond(A) how close is A to a singular matrix

II Escuela de Optica Biomedica, Puebla, 2011 SVD Solution this is obtained minimizing the error Fundamental property of SVD

II Escuela de Optica Biomedica, Puebla, 2011 SVD U and V are respectively mxm and nxn unitary matrices and is a diagonal whose elements are the singular value of the matrix A. is a mxn matrix, the singular values of A are written along the main diagonal often in descending order. The columns of U are eigenvectors (left singular vectors) of AA T and the columns of V are eigenvectors of A T A (right singular vectors).